In the Floquet picture, the Volkov state corresponds to the superposition state of the electron dressed by \(s\)  Floquet photons, i.e., \(\Psi ^{\mathrm {V}}_p(x)=\sum _{s=-\infty }^{+\infty }{F_s \mathrm {exp}\lbrace -i(q\cdot x+s k\cdot x)}\rbrace u(p)\)  (the expression of \(F_s\)  can be found in SI). The electron in the Volkov state can spontaneously decay into another Volkov state with a different quasi-momentum and thus radiate photons, the amplitude of which can be evaluated by the Feynman diagrams (see details in section 2 of SI). Especially, the entangled photon pair emission is equivalent to non-perturbative double Compton scattering of the quasi-EM wave from the electron in the EF [1]}, [2]}, which satisfies the four-momentum conservation \(q_\mathrm {f}+k_1+k_2-q_\mathrm {i}-nk=0\)  (see Fig. REF c). \(k_1\)  and \(k_2\)  are the momentum of entangled photons, and \(n\)  corresponds to the number of Floquet photons absorbed. As the energies of photons (\(k^0,k_1^0,k_2^0\) ) are much less than the electron mass (\(m\) ) and the quasi-momenta (\(q_{\mathrm {i}},q_{\mathrm {f}}\) ) in the EF when \(n\)  is reasonably small, the four-momentum conservation leads to \(k_1^0+k_2^0\simeq nk^0\) . We focus on the representative case of \(n=1\) , which corresponds to photon pairs with total frequency of \(\omega _{\mathrm {fd}}\)  in the lab frame, and can be distinguished from other cases with different \(n\)  by detecting the emitted photon pair energies (see details in section 3 of SI).
